fundamentals of acoustics(13)
TRANSCRIPT
-
8/7/2019 Fundamentals of Acoustics(13)
1/18
Fundamentals ofFundamentals ofAcousticsAcoustics
Some General WaveSome General Wave
PhenomenaPhenomena
-
8/7/2019 Fundamentals of Acoustics(13)
2/18
-
8/7/2019 Fundamentals of Acoustics(13)
3/18
This expression has two parts: first partThis expression has two parts: first part
is a standing wave field.is a standing wave field. 2 cosj t
iA p kxe[
It describes a waveform that does notIt describes a waveform that does notpropagate along the x direction, instead , thepropagate along the x direction, instead , thewaveform remains stationary.waveform remains stationary.
Such a wave is called a standing wave and isSuch a wave is called a standing wave and ismathematically characterized by an amplitudemathematically characterized by an amplitudethat depends on the position along the xthat depends on the position along the xdirection.direction.
,Tnkx ! ( 1, 2, )2x n nP! ! L
The positions of maximum pressure are
called antinodes of standing wave
-
8/7/2019 Fundamentals of Acoustics(13)
4/18
Where the pressure is zero at all time ,Where the pressure is zero at all time ,calledcalled nodes of standing wave.nodes of standing wave.
,
2
)12(T
! nkx ),2,1(
4
)12( .!! nnxP
when
The interference of sound waves
If sound waves of the same frequencyIf sound waves of the same frequencyand amplitude are superposed, theyand amplitude are superposed, theyeither neutralize or reinforce eacheither neutralize or reinforce eachotherothers effects. The phenomenon iss effects. The phenomenon is
described asdescribed as interferenceinterference
-
8/7/2019 Fundamentals of Acoustics(13)
5/18
This could be the case for two signalsThis could be the case for two signalsboth deriving from the same source,both deriving from the same source,
such as two speakers both being drivensuch as two speakers both being drivenby the same signal generator or lightby the same signal generator or lightfrom a single laser beam being split andfrom a single laser beam being split and
recombinedrecombined
1 1 1
2 2 2
cos( );
cos( ).
A
A
p p t
p p t
[ N
[ N
!
! 12 NN] !
Two signals that have a definite fixed relative
phase relation are called coherent.
-
8/7/2019 Fundamentals of Acoustics(13)
6/18
1 2 1 1 2 2cos( ) cos( )
cos( )
A A
A
p p p p t p t
p t
[ N [ N
[ N
! !
!
!
!
.coscossinsintan
),cos(2
2211
22111
1211
2
2
2
1
2
NN NNN
NN
AA
AA
AAAAA
pppp
ppppp
,,4,2,0 .TT] ss!12 NN] !
1 2 A A A p p p!
When
Maximum cooperation
,,3, .TT] ss!When
1 2 A A A p p p! Maximum cancellation
-
8/7/2019 Fundamentals of Acoustics(13)
7/18
33--9 Spherical Acoustic9 Spherical Acoustic
WavesWavesA disturbance is produced by a pointA disturbance is produced by a point
source and propagated away from thesource and propagated away from the
sphere uniformly in all direction assphere uniformly in all direction asspherical wavesspherical waves, we have spherical, we have sphericalacoustic wavesacoustic waves
Expressed in spherical coordinates theExpressed in spherical coordinates thewave equation iswave equation is2 2
2 2
02 2 2 2 2 2
1 1 1[ ( ) (sin ) ]
sin sin
p p p pc r
t r r r r r U
U U U U Nx x x x x x
!
x x x x x x
-
8/7/2019 Fundamentals of Acoustics(13)
8/18
oy
x
z
U r
N
If the waves havespherical symmetry, theacoustic pressure p is afunction of radial
distance and time butnot of the angularcoordinates UN,
-
8/7/2019 Fundamentals of Acoustics(13)
9/18
Spherical acoustic waves do not changeSpherical acoustic waves do not changeshape as they spread out. Although theshape as they spread out. Although thewavefront of spherical acoustic waveswavefront of spherical acoustic waves
can be assumed plane at great distancescan be assumed plane at great distancesfrom the source, many acousticalfrom the source, many acousticalproblems are concerned with divergingproblems are concerned with diverging
spherical acoustic waves radiated from aspherical acoustic waves radiated from asimple source rather than plane acousticsimple source rather than plane acousticwaves.waves.
-
8/7/2019 Fundamentals of Acoustics(13)
10/18
In the case of spherical symmetry,In the case of spherical symmetry,
the wave equation simplifies tothe wave equation simplifies to
)](1
[2
2
2
02
2
r
pr
rrc
t
p
xx
xx
!
xx
2 22
02 2
2( )
p p pc
t r r r
x x x!
x x x
Xpr!
Rewriting the wave equation2 2
2
02 2
X Xc
t r
x x!
x x
-
8/7/2019 Fundamentals of Acoustics(13)
11/18
The equation is of the same form asThe equation is of the same form asthe plane wave equation with thethe plane wave equation with thegeneral solutiongeneral solution
The first term represents a spherical waveThe first term represents a spherical wavediverging from a point source at the origindiverging from a point source at the originwith speed cwith speed c00; the second term represents a; the second term represents awave converging on the originwave converging on the origin..
1 0 2 0( / ) ( / )X f t r c f t r c!
1 20 0( ) ( )
r rrp f t f tc c!
1 2
0 0
1 1( , ) ( ) ( )
r r p r t f t f t
r c r c!
-
8/7/2019 Fundamentals of Acoustics(13)
12/18
The most important diverging sphericalThe most important diverging sphericalwaves are harmonic. Such waves arewaves are harmonic. Such waves arerepresented in complex form byrepresented in complex form by
The wave diminish in amplitude as theThe wave diminish in amplitude as the
distance from the source increase.distance from the source increase.
)/(1
01 crtfr
p !
( ) j t kr Ap er
[
!
The converging wave has little application inacoustics while the diverging wave is
frequently produced by a small source andhas many uses.
-
8/7/2019 Fundamentals of Acoustics(13)
13/18
The acoustic impedance ofThe acoustic impedance ofspherical wavesspherical waves
Form the equation of motionForm the equation of motion
r
p
t
u
x
x!
x
x
0
1
V xx
! d
tr
p
u 0
1
V
( )
2
0 0
1(1 ) j t kr
A jkru jkr e p
j r j r
[
[V [V
! !
It is apparent that , in contrast with planewaves, the particle velocity is not in phase
with the pressure
-
8/7/2019 Fundamentals of Acoustics(13)
14/18
For acoustic impedanceFor acoustic impedancea
Z
pu !
2
0 0 0 0
2 2
( )
1 1 ( ) 1 ( )a
j r c kr r Z j
jkr kr kr
[ V V V[! !
Za will be found to be complex
a a a a aZ r jx r j m[! !
2
2
00
)(1
)(
kr
krcr
a !
V
2
00
)(1
)(
kr
krcx
a !
V2
0
)(1 kr
rm
a
!V
Where ra
is called the acoustic resistance and Xa
theacoustic reactance
-
8/7/2019 Fundamentals of Acoustics(13)
15/18
j
a a Z Z e N!
0 0
2
1, tan
1 ( )
a
a
a
c kr xZ
r krkr
VN! ! !
0 0
2cos , cos
1a
krZ ckr
V N N ! !
N1
kr
2 21 k r
N
1
kr
2 2k rA geometrical representation ofN
is given in Fig.
-
8/7/2019 Fundamentals of Acoustics(13)
16/18
0 1 2 3 4 5 6 7 0
0 .2
0 .4
0 .6
0 .8
1
kr
30o
60 o
90o
N0 0
ar
cV
0 0
ax
cV
0 0
ar
cV
0 0
ax
cV
N
When the distance from the source is only asmall fraction of a wavelength, the phasedifference between the complex pressureand particle speed is large
1, 0, 0, / 2a a
kr r x N Tp p p=
-
8/7/2019 Fundamentals of Acoustics(13)
17/18
When kr=1, both the acousticWhen kr=1, both the acoustic
resistance and reactance areresistance and reactance areequal toequal to
And the acoustic reactance has itsAnd the acoustic reactance has its
maximum value.maximum value. WhenWhen
0
0 0/ 2, 45c andV N !
0 01, , 0, 0
a akr r c xV Np p p?
At distances corresponding to a
considerable number of wavelengths, pand u are very nearly in phase, and thespherical wave then assumes the
characteristics of a plane wave.
-
8/7/2019 Fundamentals of Acoustics(13)
18/18
This behavior is to be expected , sinceThis behavior is to be expected , since
the wave fronts of all spherical wavesthe wave fronts of all spherical wavesbecome essentially plane at greatbecome essentially plane at greatdistances from their source.distances from their source.